Problem: Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{r^2 - 36}{r^2 - 10r + 24}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 - 36}{r^2 - 10r + 24} = \dfrac{(r + 6)(r - 6)}{(r - 4)(r - 6)} $ Notice that the term $(r - 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r - 6)$ gives: $k = \dfrac{r + 6}{r - 4}$ Since we divided by $(r - 6)$, $r \neq 6$. $k = \dfrac{r + 6}{r - 4}; \space r \neq 6$